Arc approximation property and confluence of induced mappings
We say that a continuum \(X\) has the arc approximation property if every subcontinuum \(K\) of \(X\) is the limit of a sequence of arcwise connected subcontinua of \(X\) all containing a fixed point of \(K\). This property is applied to exhibit a class of continua \(Y\) such that confluence of a mapping \(f:X\to Y\) implies confluence of the induced mapping \(2^f: 2^X\to 2^Y\) and \(C(f): C(X)\to C(Y)\). Actually, the author proves in Theorem 4.4. that if \(f:X\to Y\) is confluent and \(2^Y\) (resp. \(C(Y)\)) has the arc approximation property, then the induced mapping \(2^f\) (resp. \(C(f)\)) is confluent. As a corollary he takes the following result: If \(f:X\to Y\) is confluent and \(Y\) is locally connected, then \(2^f\) and \(C(f)\) are confluent. Moreover, he proves somehow inverse results. One of them is the following: If \(2^f\) of \(C(f)\) is surjective and joining then \(f\) is confluent. In the sequel he proves similar results for weakly confluent mappings.
scientific article; zbMATH DE number 1286530
scientific article